Question

A survey of 2,150 adults reported that 53% watch news videos. Complete parts (a) through (c) below. a. Suppose that yotakeaample of 50 adults. If the population proportion of adults who watch news videos is 0.53, what is the probability that fewer than half in your sample will watch news videos? The probability is that fewer than half of the adults in the sample will watch news videos. (Round to four decimal places as needed.) b. Suppose that you take a sample of 250 adults. If the population proportion of adults who watch news videos is 0.53, what is the probability that fewer than half in your sample will watch news videos? The probability ishat fewer than half of the adults in the sample will watch news videos. Round to four decimal places as needed.) c. Discuss the effect of sarmple size on the sampling distribution of the proportion in general and the effect on the probabilities in parts (a) and (b). Choose the correct answer below. 2 OA. Increasing the sample size by a factor of 5 increases the standard error by a factor of 5. This causes the sampling distribution of the proportion to become more concentrated around the true population proportion of 0.53 and increases B. Increasing the sample size by a factor of 5 decreases the standard error by a factor of 5. This causes the sampling distribution of the proportion to become more concentrated around the true population proportion of 0.53 and C. The probabilities in parts (a) and (b) are the same. Increasing the sample size does not change the sampling distribution of the proportion. 2 the probability in part (b). decreases the probability in part (b)

Answer 1

Given: p 0.53, q 1-p 0.47,n-50 Taking normal approximation to binomial, the sampling distribution of sample proportion follows normal distribution Shape: Normal p ~ N(μ, σ2), where mean, = p = 0.53 p 0.53 variance, σ2-px (1-p)--0.53 × 0.47 standard deviation, σρ-Vg-- - 0.004982 50 0.53 x 0.47 50 0.070583 ap 0.070583 p~N(0.53,0.0705832) A 〈 0.5) Normal Distribution, μ = 0.53, σ = 0.070583, P(ρ < 0.5) = Area to the left of we convert this to standard normal using z- 0.5 0.5 (0.53) 0.070583-0.42503 0.43

0.3336 0.43 P(p< 0.5)-P(Z< -0.43) 0.3336 (from z-table) P(2-0.43): in a z-table having area to the left of z, locate -0.4 in the left most column Move across the row to the right under column 0.03 and get value 0.3336 P(P < 0.5)-0.3336-33.36% ® n= 250, p= 0.53 mean of sampling distribution, Hp p 0.53 Mp = 0.53 px(1-p)-0.53x 0.47 - 0.0009964 variance, σ-- 250 0.53 x 0.47 250 Vơ2- standard deviation of sampling distribution, σρ- 0.031566 0.031566 Normal Distribution, μ = 0.53, σ = 0.031566, P(þ < 0.5) = Area to the left of 0.5

we convert this to standard normal using z- 0.5 (0.53) z = 0.031566 -0.950396 -0.95 0.95 P(ρ < 0.5) = P(Z <-0.95) = 0.1711 (from z-table) P(Z-0.95): in a z-table having area to the left of z, locate -0.9 in the left most column Move across the row to the right under column 0.05 and get value 0.1711 P(p < 0.5) 0.1711 = 17.11% c) Since the sample size is 5 times, the standard error reduces by a factor of v5 times, this causes the sampling distribution to concentrate more around mean decreasing the probability in (b). Option B) oThank youes

Answer 2

A survey of 2,250 adults found that 56% got news both online and offline

during a typical day.

a. Suppose that you take a sample of 50 adults. If the population

proportion of adults who get news both online and offline in a typical

day is 0.56, what is the probability that fewer than half in your sample

will get news both online and offline in a typical day?

b. Suppose that you take a sample of 250 adults. If the population

proportion of adults who get news both online and offline in a typical

day is 0.56, what is the probability that fewer than half in your sample

will get news both online and offline in a typical day?

c. Discuss the effect of sample size on the sampling distribution of the

proportion in general and the effect on the probabilities in parts (a)

and (b).